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Full text of "Notes on the design of latticed columns subject to lateral loads"

TECHNICAL NOTES 
NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS. 



No. 98. 



NOTES ON THE DESIGN OF LATTICED .COLUMNS SUBJECT TO LATERAL LOADS- 

By Charles J. McCarthy, 
Bureau- of Aeronautics-, Navy Department. 



May, 1S22. 



NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS. 



TECHNICAL NOTE NO. 98. 



NOTES ON THE DESIGN OF LATTICED COLUMNS 
SUBJECT TO LATERAL LOADS. 

By diaries J. McCarthy, 
Bureau of Aeronautics, Navy Department. 

. ; _ The increasing interest in the use of metal for the con- 
struction of aircraft makes timely a discussion of the problems 
and difficulties to be met in the design of efficient compression 
members. No rational column formula has yet been developed 
which gives results which are sufficiently precise for the de- 
sign of airplane members, and consequently it is necessary to 
fall back upon experimental testing. In order to derive the max- 
imum benefit from experiments, however, it is necessary that the 
experiments be guided by theory, and it is the object of this 
paper to suggest a method of procedure by means of which the data 
needed to modify existing formulae may be obtained with a minimum 
of tests. 

Although it is common in wing construction to find v-ing beams 
continuous over several supports, for the sake of simplicity this 
discussion will be limited to that of a simple column supported 
at both ends and subjected to uniformly distributed loads perpen- 
dicular to its axis and to end loads either axially or eccentri- 
cally applied. 



- s - 

Ideal Columns . 

The failing strength of a perfectly straight homogeneous 
column with pinned ends in whioh the compressive load is exactly 
axially applied is expressed by Euler 1 s formula: 



P* -no /K 



= TT 8 



2 



E 



("t") (1 > 



Where P' is the critical end load; 

A is the cross sectional area of the column > 
E is the modulus of elasticity of the material ; 
L is the least ratio of length to radius of . 
gyration. 



K 

It should be kept in mind that the critical load calculated 
from the aboTre formula is the end load required to buckle the 
strut and that for loads smaller than this the ideal column re- 
mains perfectly straight. It is apparent also that the column ■ 
will fail elastically as soon as the stress at the ends reaches 
the elastic limit of the material. Consequently the curve of ul- 
timate stress vs. L/K for ai Euler column has the form of the 
right hand curve of Fig. 1. ^ 

If now instead of being axially applied, the end load has- an 
eccentricity, h, bending stresses are introduced which increase 
the stresses in the fibers of the column and decrease the magni- 
tude of the load which will cause failure* In the case of a prac- 
tical strut, variations in the shape and .thickness of the section, 
initial curvature and other imperfections have the effect of giving 
an eccentricity to the- end load. 



- 3 - 

The equation for the maximum intensity of stress under these 
conditions is given "by Morley as* 



■* - A ( 1 + 2K : 



sec 



2 J ei ; 



(2) 



Where P is the end load applied; 

A 'is the cross sectional area of the column; 
h is the eccentricity, i. e. , the distance from 

the point of application of the load to the 

centroid of the section; 
d is the depth of the section in the plane of 

bending; 
k is the radius of gyration in the plane of 

bending. 

This formula may be expressed (approximately) as follows: 



f + = ~ 



1 + 



1. 2 hd " 
2K a 

i -siL 

TT 2 EI 



(3) 



which may be simplified by substituting the Euler load, P' for 
the expression — —■. 



Thus, f t = f 



+ P (-21— \ (1 

a vp' - py v 



.6 hd ^ 
K 3 J 



(4) 



Failure occurs when f^ reaches the elastic limit of the 



material in compression, f c 

;ed in equ 
approaches infinity 



It will be noted in equation (4) that as P approaches P' 
P r 



the ratio 

P' - P 

The curves of Fig- 1, which are' taken from Morley, are of in- 

terest as they show how end stress at failure is affected by vary- 
. * Morley. Strength of Materials 1916, p. 376. 
£* Ibid. p. 376. 



_ 4 - 

ing eccentricities and varying values of L/K. 

Another condition to be considered is the combination of the 
axial loads with forces perpendicular to the axis of the strut. ■ 
The deflection of the strut which is produced by the lateral 
loads has the effect of making the axial loads eccentric with a 
consequent increase in the maximum bending moment in the strut. 
The total bending moment is the sum of an infinite series , the 
first two terms of which are the bending moment due to the later- 
al load, and that of the product of the axial load by the deflec- 
tion of the column under the lateral loads. For a uniformly dis- 
tributed lateral load of w per unit length the exact equation 
for the maximum bending moment at the center of the column, M 
under the combined loading, is given by 



This may be more conveniently expressed by Perry 1 s approxi- 
mate formula: 

where M is the maximum bending moment due to the lateral loads 
alone, and the other symbols have the same significance as before. 

If Z equals the section modulus the maximum fiber stress 
due to bending equals 



*' Morley. p. 283. 



4 

The error introduced by this approximation amounts to less 
than 3 percent for ratios of p top 1 up to 0« 9. 

Combining equations (4) and (7) results in a general formula 
for the maximum intensity of stress in a perfectly straight col- 
umn of homogeneous material with pin ends, loaded with a unif orin- 
ly distributed transverse load, which, acting alone, would pro- 
duce a maximum bending moment M; and in addition, an end load, 
P which is applied a distance, h from the centroid of the 
section in the plane of bending. 

% a . a vp' - p ; v k 3 ; z p> - p K J 

This formula is an approximation, but is sufficiently precise 
when the ratio of p to P r does not exceed 0.9. For higher val- 
ues the formulae of equations (2) and (5) are recommended. It 
should be noted here that p« in equation (8). is introduced 

merely as a substitute for the expression n I 1 and its value 

L 

is not limited by the strength of the material at the elastic 
limit, as is the case when calculating the strength of a "Euler" 
strut, as has been explained in connection with equation (1). 

Failure of the column may be expected to occur when the total 
fiber stress f t reaches the elastic limit of the material in 
compression. 
Latticed Golumns . 

The above formula, equation (8)/ has been derived for a col- 
umn of homogeneous material, but may be applied to one built up 



- 6 - 

of longitudinal members 01 flanges which, are laced together witL: 
lattice bars if attention is paid to the fact that the individual 
flanges act independently as little columns of length equal to 
,the lattice spacing. The maximum fiber stress of equation (8) 
should be limited to the end stress which the flange will carry- 
as a pin-ended column whose length equals the lattice spacing. 
It is not correct to base the design of a lattice column on the 
assumption that the column is homogeneous and then limit the 
spacing of the lattices such that the — of each flange between 

K. 
the points of attachment of lattices does not exceed the j? of 

the column as a whole. This procedure leaves no margin to allow 

for the increase in stress in the flange due to its acting as an 

independent column between lattices. 

Another point to be noted is that when the column is acting 

as a beam the flanges receive their load from the lattices', and 

the flange as a whole acts approximately along its centroidal 

axis, in calculating the section modulus, Z, therefore, it 

will be. more nearly representative of the true" condition if the 

"extreme fiber distance"', Y, is measured from the centroid of 

the flange instead of taking one-half the depth of the column. 

This amounts practically to assuming that the stress is uniformly 

distributed over the flange section. 

Applicat ion of "Theory to Practical Columns . 

Many attempts have been made to develop a rational formula 

which will properly express the state of stress in a practical 



- 7 - 

column, bat this has not yet been accomplished. Paaswell says * 
in commenting on a recent paper on the subject: "Briefly, a col- 
umn is an engineering structure subjected to a compressive force 
of a determinate character and to a flexure absolutely indetermi- 
nate and unpredictive with any mathematical certainty. This of 
course refers to columns presumably axially loaded. The intro- 
duction of flexural stresses occurs in a manner which can only 
form a matter of conjecture." 

Chew** classifies imperfections which may reduce the strength 
of an actual column as follows: 

"1. Initial stresses in material due to manufacture. 

2. Variation in strength of component parts of section. 

3. Crookedness of component parts. 

4. Crookedness of whole member. 

5. Local stresses due to details and shop work. 

6. Accidental eccentricity. 

7. Deflection caused by the foregoing imperfections. « 
Basquin*** too has gone into the problem of developing a 

formula for the design of columns which will take separate ac- 
count of the stresses to be anticipated in the actual column due 
to crookedness, probable eccentrioities, etc. , but the tests on 
which his work has been based were not extensive enough to warrant 
the general application of his conclusions to design. 

It has been found, furthermore, that a built-up column as re- 
gards bending action does not act as a perfect unit* Fig. 3 is 

* Proc. ASCE, January, 1923. 

** Proc. Am. Soc. Civil Engrs. ," May", 1911. 

*** Basquin on Columns Journal W. S.C.E. 3 1911. 



- 8 - 

taken from the comments of Prof. H. F. Moore, of the University 
of Illinois*, and gives the results of a series of tests conduct- 
ed at the University of Illinois to determine the ratio of com- 
puted to actual fiber stress in the cross section of members 
built up of channels, fastened together with different types of 
lacing. Quoting Prof. Moore, "Short column sections (all of the 
same length) were tested as beams with flexure in a plane paral- 
lel to the plane of the lacing. Assuming integrity of action of 
cross section, the extreme fiber stresses in a test beam were 
calculated for various loads, and the actual fiber deformations 
developed under these loads were measured by means of a strain 
gauge, and the actual fiber stresses, determined from the ob- 
served elongations and compressions, were indicated by the strain 
gauge. In Fig. 3 is shown the variation of f lexural efficiency 
with computed fiber stress for various column sections. In a 
column of usual length in structures (^ = 50 to 75), the com- 
pressive stress is the principal stress in the column and the 
flexural stress is not very high; so in comparing the flexural 
efficiencies of different column sections the efficiencies under 
low flexural stresses are most significant. The superiority of 
the double-laced section with rivets at the crossing of the bars 
is evident; the efficiency of this section at low stress proved 
to be the same as the efficiency of a pair of channels tested in ' 
flexure in a plane parallel to the plane of their webs. The low 

efficiency of channels connected by mea ns of batten plates is 
* Illinois University Bulletin No. 40. 



- 9 - 

noteworthy as is the very low efficiency of two channels connect- 
ed by no n- overlapping bars with only one rivet for each end of a 
bar. In each test piece approximately the same weight of lacing 
material was used, .and all tests were in duplicate. Each test 
was loaded symmetrically at two points of the span, and the spans 
were the same for all test pieces* n 

Major Nicholson has also observed, in a series of tests on 
metal girders designed for airplanes, that the deflections of 
latticed girders under transverse loading exceeded those of simi- 
lar girders with solid webs.* 

The weight of other authorities whose opinions are in the 
same vein might be added, .but those quoted^ above should be suffi- 
cient to indicate the difficulties to be encountered in attempt- 
ing to calculate the distribution of stress in compression mem- 
bers. In ordinary structural design these difficulties are some- 
times circumvented by the device of limiting the calculated max- 
imum intensity of stress due to the combination of end and side 
loads to the allowable end stress on the strut as a simple pin- 
ended column. This procedure is illustrated in the design of a 
large derrick boom, which has been worked out in detail by M. G. 
Bland in a paper entitled "Investigation of Stresses in Derricks?*'" 
This procedure is conservative, and while it probably gives re- 
sults which are quite satisfactory for structural work where a 
slight excess in the weight of a member is not a serious matter, 

it is not sufficiently precise for general use in the design of 

* The Development of Metal Construction in "Aircraft Engineer- 
ing," London, March 12, 1920. 
** Trans. ASGE, 1920. 



- 10 - 

airplane girders, particularly when the end load is relatively 
small compared with the transverse load. As the magnitude of the 
end load approaches aero, the column becomes a siicple beam, but 
according to the above method the criterion for the maximum in- 
tensity of fiber stress is still the limiting stress on the mem- 
ber as a pin-ended column. 

We are thus forced to the conclusion that for the design of 
compression members the theoretical formulae must be reinforced 
and modified by experiments on the particular type of column 
which is to be used. The most hopeful procedure is to select a 
formula such as equation (8) and by a series of careful experi- 
ments on full-size columns, determine the factors which must be 
introduced into this formula to make it fit the actual members. 
•Referring to equation (8), it will be noted that there are two 
quantities, P' and h, to which modifying factors could be 
applied. 

As has been stated previously, P 1 in this formula is mere- 
ly a shorthand expression of the quantity — — 5-. Now the only 
quantity in this expression' to- be determined experimentally is 
the E, which represents the modulus of elasticity of the built- 
up member. This can be easily found by measuring the deflection 
of the column when loaded as. a simple beam by a. transverse load 
concentrated at the center, and solving for E in the well-known 

deflection formula S «= ~ ?£-. The procedure may be improved, 

48 EI 

howeveT, by retaining ~E as the modulus of elasticity of the 



- 11 - 

material of which the column is built and introducing a coeffi- 
cient C into the formula: thus S = i -?fe ■ may oe looked 

48 CEI 

upon as the "form factor" for the section,, and represents the 
ratio of the stiffness of the actual column to that of a solid 
theoretical column of the same material. This coefficient could 
then be applied to the calculation of P 1 , but it will be pref- 
erable to introduce into equation (8) and use the modulus of 
elasticity of the material in calculating p». 

The term h, may be considered as being the sum of the known 
eccentricity of the application of the load to ends of the col- 
umn H, and an equivalent eccentricity which represents the over- 
all "constructional" eccentricity of the actual column, that is, 
the sum of the imperfections of the actual strut and is designat- 
ed by e. To find e, it is necessary to build and test as pin- 
ended struts with axial loads, a number of full-sized columns of 
varying lengths of the type to be used. These test specimens 
should of course be built as far as possible to the same quality 
of workmanship and straightness as will be followed in the con- 
struction of the columns or beams to be used in the airplane it- 
self, a column formula may be plotted from the results of these 
tests and e calculated from the relation 



0-6 ed = /+ Jl - i\ / CP 1 - Pp * 
K 



-- e^- 1 ) r^) < 9 > 



Where f c = the compressive elastic limit of the mater- 
_ ial for homogeneous struts and for latticed 
struts the limiting unit end stress on an 
individual flange of a length equal to the 

lattice spacing, 



* From ifoxley, p. 376. 



- 13 - 

. — - = the observed ultimate end stress as a pm- 
ended column. 

p* = critical end load calculated from Euler' s 
formula. 

G = the form factor coefficient' mentioned above 
.(This factor does not appear in the formula, 
as given by Morley. ) 

With latticed columns additional tests must be made of the 
strength of the individual flanges as pin-ended columns to deter- 
mine the proper value of f to be used in the above formula. 

The above equation (9) appears somewhat formidable , but it 

will be found from experiment in most cases that the eccentricity 

e, can be sufficiently expressed as a simple function of the 

or 
length/of the • L/K of the column. 

Introducing the above modifications, equation (8) may be re- 
written as 

* _ Z + Z / CP f \ / 0.6 (e + H) d N M / GP' \. (10) 

° " a a vgp» - p ; \ k 5 ; . z \cp' - py 

Forces in the Bracing of Latticed Q-olumns . 

■ The forces which act upon lattice bars have been divided by 
Basquin into three classes:* "First, those introduced in the fabri- 
cation of the column;; second, those due to transverse shear caus- 
ed by local bends in the column; and third, those due to trans- 
verse shear caused by general inclination of the column. " The 
latter two conditions have been investigated in a series of care- 
ful extensometer tests by Talbot and Moore.**' In case of a 
column built of two channels latticed together- with flat bars and 

* journal W.S.C.E. , 1913, p.493. 

** "An Investigation of Bui It -tto Columns Under Load," University 
of Illinois Bulletin #40 of June 10th. 



*■ 13 - 

with an average end stress of 10,000 pounds per square inch, 
they conclude that:. "It is evident from the tests that the rel- 
ative stress in the two-channel members varies considerably from 
end to end and that the stress in the lattice bars also varies. 
It seems probable that the transverse shear developed may be 

* 

traced largely to irregularities in outline, or at least that 
these irregularities may be expected to cover up other causes of 
stress in the lacing of centrally- loaded columns, if we include 
in such irregularities all unknown eccentricity. The futility 
o£ attempting to determine analytically the stresses in column 
lacing, using as a basis either a bending moment curve which var- 
ies from end to middle or an assumed deflection curve, is appar- 
ent from a study of the variation of stress in the columns of the 
tests and in that of the lattice bars. " 

It is nebessary, nevertheless, to find some means of approxi- 
mating the loads in the lattice members. The method most favored 
in the design of structural columns is to assume that the column 
is loaded with a uniformly distributed transverse load w, where 
w is the transverse load, which, considering the oolumn as a sim- 
ple beam, will produce a maximum fiber stress equal to the differ- 
ence between the elastic limit of the material and the end unit 

stress allowed by the column formula. The vertical component of 

wl 
the load in the lattices at the ends of the column equals -r- , 

which is assumed to be equally distributed between the lattices 
cut by a vertical pl ane normal to the axis of the girder-* 
* Spofford - "The Theory of Structures,"' 1915, p. 303. 



- 14 - 

Alexander* has investigated the distribution of the shearing 
stresses in an ideal column, using equations which involve the 
true elastic curve of flexure of the column. His expression for 
the maximum shear may be put into the form 



R 



_ IT Z / ^ P 



(* -^) CU) 



Where R = the shear at end of column; 

Z = section modulus; 

fo= limiting stress on short column; 

P'= Euler .crippling load. 

The constant rr in the above equation is increased to 5 for 
actual struts to allow for longitudinal irregularities and slight 
imperfections in fitting and securing the lattice bars. 

It may be noted that the assumption that the shear may be 
determined on the basis of the uniformly distributed lateral load 
mentioned above amounts to assuming a parabolic curve for the de- 
flection of the column. This latter approximation gives a value 
of 



■a - M 

R ~ L 



( f c-i) 



which exoeeds the shear calculated by the more exact method but 
is less than that recommended by Alexander for practical columns. 
Until this subject has been more thoroughly investigated by exper- 
iment, it is recommended that the shear in lattice bars be calcu- 
lated by Alexander's formula, 



R L 



( f °-T) • ^ 



* TSta. Alexander, "ffolmms and Struts," 1912, Chap. X. 



-lo- 
in view also of the approximate nature of this method., and the 
variations in shear due to local irregularities, etc. , no attempt 
should be made to* vary the strength of the latticing along the 
length of the column. Alexander points out that "the points 
where the deflection is a maximum and the shearing forces nil, 
are unknown and certain to be different in each strut," and con- 
cludes^ "each case must be considered on its own merits. No gen- 
eral formula can be given for even the probable limits of reduc- 
tion in shearing stresses. n 

/ P ! \ 
It will be noted that the expression / f c - -r- j in equa- 
tion (13) would equal the fiber stress in an ideal strut due to 

flexure. To apply this formula to a practical- strut, substitute 

P' 
for — the fiber stress permitted by the experimental formula. 

The resulting shear in the lattices will be 20 per cent in excess 
of that determined by the procedure given by spofford. 

To find the shear in the lattice bars of a strut under com- 
bined end and transverse loads, let the sum of the second and 
third terms of equation (10) equal f^ 

Then R = ~ (f b ) (13) 

This use of this formula is recommended. 
Illustrative Problem .. 

To illustrate the application of this method of analysis an 
example will be worked out. The column chosen is of suitable 
proportions to be used as a portion of the wing beam of a large 



- 16 - 

airplane. Its strength about the horizontal axis only will be in- 
vestigated. Fig. 3 is a sketch of the column. It will be noted 
that the latticing on the top and bottom faces is entirely inside 
the flanges. While not the best design from a structural stand- 
point, it is desirable to facilitate sliding the wing ribs along 
£he beam in the assembly of the wing panel. 

.Let P = the total end load including the factor of safety = 
30,000 lbs. 

M = maximum bending moment due to the uniformly dis- 
tributed transverse loads = 81,600 inch pounds. 

L = length of beam between points of inflexion = 144 
inches. 

A = area of flanges = 0.88 square inches. 

I = moment of inertia - 6.9 (inches)* 

K = radius of gyration of section about axis XX = 3.8 
inches. 
L/K = 144/2.8 = 51. 

y = distance from centroid of section to centroid of 

flange =2.8 inches. 
Z = Section modulus = I/y = 2.46 inches cubed. 

f c = crippling end stress on one angle of flanges as a 
pin-ended column whose length equals the pitch of 
the lattices = 105,000 lbs. per sq. in. 

TT3EI 

P = failing load as a pin-ended column. 

In the absence of experimental data on columns of this type, 
curve A-of"«--Fi 

curve A of Fig. 4 has been more or less arbitrarily Chosen to rep- 
resent the relation between L/K and the failing end stress. 

G - the form factor coefficient has been assumed equal 
to 0.8. 

Then ■ 5 2 -=' 58,000 lbs. per sq. in. from Fig. 4. 
and OP' = 6.8^(30,000,000) (0,88x3.8°) = , 79 , 000 . 

(144) 2 



- 17 - 



Find the "constructional eccentricity" of the column e fi'on 
equation (9). 



0. 6 e 



J[d) = / 105000 _ X N / 79000 - 58000 (. 88 )\ = 0# 2 9 

(K) 3 V 58000 / V 79000 / 

and e = 0.606" which is -^=z. of the length of the column. 

Substituting the above value of u into equation (10) 

the maximum fiber stress in the flanges at a point of attachment 
of the lattice equals 

= £ 2 / CP' \ / 0.6 ed \ + M / GP 1 ^ — 

* A + A VCP 1 - P J V K 3 J Z VCP' - P y^_ 

34100 + 34100 (™™°) (0..29) + 23^0 / 79000 \ = 

44.LUU.+ 44AUU V 49000^ .2.46 V 49000 / 

= 103400 lbs. per sq. in. 

Since f-^ is less than 105000 lbs. per sq. in, , the area 
provided in the flanges is sufficient. 
Load in Lattice Members. 



By equation (13) the shear at the end of the column equals 

5Z 
L 



R = ~ (* b ). 



There f-^ equals the maximum flexual fiber stress as ex- 
pressed by the second and third terms of equation (8). 

r = 5 jjjh 46 ) 69300 = 5920 lbs. 
. 144 

Assuming the above shear distributed equally between the four 
lattices cut by a plane perpendicular to the longitudinal axis of 



~ 18 - 

the column, the total load in each lattice equals 

5920 x 7,25 = 19Q5 lbfl# 
4 5-62 

Strength of Individual Lattices . 

Assume that the lattice in compression is supported at the 
center by the adjaoent lattice which is in tension and that the 
lattice fails as a pin-ended column whose length is equal to one- 
half the length of the lattice between centers of flange rivets. 

Area of section = 0.0325 sq. in. 

Least radius of gyrations = 0.075 in. 

L/K = :fM = 45 - 

In the absence of test data on lattices as used in 'this -de- 
sign a column formula somewhat more conservative than Rankin' s 
has been arbitrarily chosen. Using steel having an elastic limit 
of 100,000 lbs. per sq. in. , the allowable p/A for an L/K of 45 
equals 60,000 lbs. per sq.in. . . 

The strength of the lattice = 60,000 x 0.0325 = 1950 lbs., 
which exceeds the required strength of 1905 lbs. The lattice de- 
sign is therefore satisfactory. 



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Mild steel strut of circular 



Fig. 1. 



Horley- strength of mater- 
ials-1916. 



Specimen with webs of channels 
vertical. 



43 

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3000 4000 6000 8000 10000 13000 





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mm 



Computed fibre stress in flexure-lbs. /sq, in. 
Efficiency=( computed fibre stress) -f (observed 

fibre stress) 
Fig. 2. 






Sect ion- "A-A" 

K i|.H 



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280,000 



240,000 



300,000 



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